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I just have found out these information in my presentation work:

a) $V\subset\mathbb R^n $ open, $D:=\{(x,x)|x\in\mathbb R^n\}$ and a function $\alpha:(V\times V)\backslash D\rightarrow \mathbb R$ with:

b) $\alpha(x,y)\in C^1(V\backslash\{y\})\cap C^2(V\backslash\{y\})$ and

c) $\triangle_x\alpha(x,y)\in L^1(V)$ and

d) $\forall u\in C^1(V)\cap C^2(V): u(y)=-\int\limits_V\alpha(x,y)\triangle u(x)d^nx-\int\limits_{\partial V }u(x)\partial_{\nu_x}\alpha(x,y)d\sigma_x $

Now I want to check if $\alpha:V\backslash\{y\}\rightarrow\mathbb R$ is solving the following problem:

$\triangle\alpha=0$ in $V\backslash\{y\}$ with $\alpha=0$ on the boundary of $V$.

I have absolutely no idea how to prove this ;( I will be very happy about some help! Thank you guys!

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What you are trying to do is show that $\alpha$ is a fundamental solution (or Green's function) for the Laplace equation.

To get started, try using integration by parts / divergence theorem on the integrals in d), and see where that leads you.

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